If a solution contains `10^(-6)M` each of `X^(-),Y^(-2)` and `Z^(3-)` ions, than upon addition of `AgNO_(3)(s)` slowly to the above solution with striing `: (` Given `: K_(sp)(AgX)=9xx10^(-14),K_(sp)(Ag_(2)Y)=4.9xx10^(-21),K_(sp)(Ag_(3)Z)=5.12xx10^(-29))`

Options

A. `Ag_(3)Z` will be the first one to precipitate out.
B. `Ag_(2)Y` will be the first one to precipitate out.
C. `AgX` will be the first one to precipitate out.
D. Nothing can be said with certainity.

Explanation

To solve the problem, we need to determine which anion (X^-, Y^2-, or Z^3-) will precipitate first upon the addition of AgNO3 to the solution. We will do this by calculating the concentration of Ag^+ ions required to initiate the precipitation of each anion based on their respective solubility products (K_sp).

### Step-by-Step Solution:

1. **Identify the Anions and Their K_sp Values**:
- X^-: K_sp(AgX) = 9 × 10^(-14)
- Y^2-: K_sp(Ag2Y) = 4.9 × 10^(-21)
- Z^3-: K_sp(Ag3Z) = 5.12 × 10^(-29)

2. **Calculate the Required Ag^+ Concentration for Precipitation of X^-**:
- The solubility product expression for AgX is:
\[ K_{sp}(AgX) = [Ag^+][X^-] \]
- Rearranging gives:
\[ [Ag^+] = \frac{K_{sp}(AgX)}{[X^-]} = \frac{9 \times 10^{-14}}{10^{-6}} = 9 \times 10^{-8} \, M \]

3. **Calculate the Required Ag^+ Concentration for Precipitation of Y^2-**:
- The solubility product expression for Ag2Y is:
\[ K_{sp}(Ag2Y) = [Ag^+]^2[Y^{2-}] \]
- Rearranging gives:
\[ [Ag^+] = \sqrt{\frac{K_{sp}(Ag2Y)}{[Y^{2-}]}} = \sqrt{\frac{4.9 \times 10^{-21}}{10^{-6}}} = \sqrt{4.9 \times 10^{-15}} \approx 7 \times 10^{-8} \, M \]

4. **Calculate the Required Ag^+ Concentration for Precipitation of Z^3-**:
- The solubility product expression for Ag3Z is:
\[ K_{sp}(Ag3Z) = [Ag^+]^3[Z^{3-}] \]
- Rearranging gives:
\[ [Ag^+] = \sqrt[3]{\frac{K_{sp}(Ag3Z)}{[Z^{3-}]}} = \sqrt[3]{\frac{5.12 \times 10^{-29}}{10^{-6}}} = \sqrt[3]{5.12 \times 10^{-23}} \approx 8 \times 10^{-8} \, M \]

5. **Compare the Required Ag^+ Concentrations**:
- For X^-: 9 × 10^(-8) M
- For Y^2-: 7 × 10^(-8) M
- For Z^3-: 8 × 10^(-8) M

6. **Determine Which Anion Precipitates First**:
- The lowest concentration of Ag^+ required is for Y^2- (7 × 10^(-8) M). Therefore, Y^2- will precipitate first.

### Final Answer:
Y^2- will precipitate first upon the addition of AgNO3.